Certainly Quinn needs to find him more than 9 mins a night.
He's useful in transition, similarly to Pouliot and Hagelin in the past as player who can move the puck across the neutral zone well and can retrieve dump-ins. He's also got good instincts as a goal scorer, he's got that kind of confidence to pounce on a half-step advantage on a turning defender and has a heavy, if not that quick, shot.
All that to me is a player who could be a useful piece of a middle six line. And particularly as a player who really could add a lot on the offensive-side of the game. It's tricky because he doesn't have the creativity or touch to be a top-six scoring threat. His contributions won't primarily be in his stat-line, which is why I'm not sure he'll be utilized correctly.
I think asking him to hit more or to try to turn him into a checker/pk-er would be a mistake. He's got a good attitude by all accounts, but it's ignoring his actual strengths and setting him up for failure because he doesn't have the awareness or the processing speed to know what to do with the puck once he's gotten it or how to anticipate his opponents as a five man unit.
I think playing with a Lafrienerre would be a great fit for him in that you've got a guy who can take control of the play in the offensive zone, directing his linemates and find them in great positions, as well as generating a lot out of chaos and scrambles which Gauthier should be able to make possible with his size and speed, and his shot first preference. But if I'm coaching Gauthier, I'm really trying to get him to see himself as a player who gets the puck into the attacking zone and frees it up for his linemates. I'd be more insistent on being stronger on his stick, improving his hand-eye to knock-down passes, and bearing down on the puck. Telling him to hit is a waste, he'll just be another Brett Howden IMO.